SOLUTION: I have a work sheet w/ some natural logs that I need help with. e^x * e^(3x) = 4 e^9 = (e^x)^3 (1/e^9) Thank you so much!

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: I have a work sheet w/ some natural logs that I need help with. e^x * e^(3x) = 4 e^9 = (e^x)^3 (1/e^9) Thank you so much!      Log On


   

Question 71077: I have a work sheet w/ some natural logs that I need help with.
e^x * e^(3x) = 4
e^9 = (e^x)^3 (1/e^9)
Thank you so much!
Found 2 solutions by Earlsdon, jim_thompson5910:
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
1) Solve for x:
%28e%5Ex%29%28e%5E%283x%29%29+=+4 Take the natural log of both sides.
ln%28%28e%5Ex%29%28e%5E%283x%29%29%29+=+ln%284%29 Apply the product rule for logarithms to the left side.
ln%28e%5Ex%29+%2B+ln%28e%5E%283x%29%29+=+ln%284%29 But ln%28e%5Ex%29+=+x
x+%2B+3x+=+ln%284%29 Simplify.
4x+=+ln%284%29 Divide both sides by 4.
x+=+%281%2F4%29ln%284%29 Evaluate using your calculator or log tables.
x+=+0.3466 Approximately.
2) Solve for x:
e%5E9+=+%28%28e%5Ex%29%5E3%29%281%2Fe%5E9%29 Multiply both sides by e%5E9
%28e%5E9%29%28e%5E9%29+=+%28e%5Ex%29%5E3 Simplify both sides.
e%5E18+=+e%5E%283x%29 Take the natural log of both sides.
ln%28e%5E18%29+=+ln%28e%5E%283x%29%29
18+=+3x Divide both sides by 3.
x+=+6

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
e%5Ex+%2A+e%5E%283x%29+=+4Start with given expression
e%5E%28x%2B3x%29+=+4The left side can be written as e%5E%28x%2B3x%29 since e%5Ex%2Ae%5Ey=e%5E%28x%2By%29
ln%28e%5E%28x%2B3x%29%29+=ln+4Take natural log of both sides, this undoes the natural base e
4x=ln+4Solve for x by dividing both sides by 4
x=%28ln+4%29%2F4=0.34657359Approximately


Do the same thing to the next problem
e%5E9+=+%28e%5Ex%29%5E3+%281%2Fe%5E9%29Start with given expression
e%5E9+=+e%5E%283%2Ax%29%2Fe%5E9The right side can be rewritten as e%5E%283x%29 since e%5Ex%5Ey=e%5E%28x%2Ay%29 (in other words e%5Ex%5E3=e%5E%283x%29)
e%5E18+=+e%5E%283%2Ax%29Multiply both side by e^9
ln%28e%5E18%29+=ln+%28e%5E%283%2Ax%29%29Take natural log of both sides, this undoes the natural base e
18+=+3%2AxSolve for x by dividing both sides by 3
x=18%2F3=6